168 
Proceedings of the Royal Society of Edinburgh. [Sess. 
Robinson, L. W. (1889): Thyagaragaiyar, V. R. (1899). 
[Question 10254. Educ. Times , xlii. p. 332 : Math, from Educ. Times , 
liv, p. 54.] 
j [Question 14240. Educ. Times , lii, p. 270: Math, from Educ. Times , 
lxxiii, p. 56.] 
Already known results. 
Rogers, L. J. (1891). 
[Question 10992. Educ. Times, xliv, p. 199.] 
This concerns the first primary minor of C(a, h, c, . . .) when a-\-b-\-c 
+ . . . = 0, the problem being to resolve the said minor into linear factors. 
No solution ever appeared. Probably the proposer had in his mind 
special circulants of the form 
C (a - b , b - c , c — d , d - a) , 
the first primary minor of which is 
a - 
b 
b- 
c 
c — d 
d- 
a 
a - 
b 
b - c 
and .* 
e — 
d 
d- 
a 
a — b 
so that its factors are factors of C(a, b, c, d). 
abed 
d a b c 
c dab 
1111 , 
Woodall, H. J. (1894). 
[Question 12426. Educ. Times, xlvii, p. 305 : Math, from Educ. Times T 
lxiii, pp. 35-36.] 
The real point of interest here concerns the product of two circulants. 
If the circulants be zero-axial, the ordinary multiplication-theorem, though 
of course producing a circulant, does not produce one that is zero-axial ; 
so that, strictly speaking, the product is in this case not of the same form 
as its factors. It is shown, however, that what is not true of the zero- 
axial circulant is true of the zero-axial circulant divided by the sum of 
its elements. Thus, by row-multiplication 
. a b 
. x y 
ax + by 
bx 
ay 
b . a 
. 
V • x 
= 
ay 
ax + by 
bx 
a b 
x y . 
bx 
ay 
ax + by 
and consequently on removing the factors a + b , x + y , (a- f- b)(x + y) from 
the three determinants respectively we have 
1 1 1 
1 1 1 
1 1 1 
b . a 
. 
y . x 
= 
ay ax + by bx 
a b 
x y . 
bx ay ax + by 
